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Let A and B be nonempty sets. Prove the following statements. (a) For any function f : A → A, if f o f is onto, then f is onto. (b) For any function f : A → B, if g : B → A is a function so that g o f = IA, then f is one-to-one.

Let A and B be nonempty sets. Prove the following statements. (a) For any function f : A → A, if f o f is onto, then f is onto. (b) For any function f : A → B, if g : B → A is a function so that g o f = IA, then f is one-to-one.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.3: Properties Of Composite Mappings (optional)
Problem 12E: Let f:AB and g:BA. Prove that f is one-to-one and onto if fg is one to-one and gf onto.
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Let A and B be nonempty sets. Prove the following statements. (a) For any function f : A → A, if f o f is onto, then f is onto. (b) For any function f : A → B, if g : B → A is a function so that go f = IA, then f is one-to-one.
Transcribed Image Text:Let A and B be nonempty sets. Prove the following statements. (a) For any function f : A → A, if f o f is onto, then f is onto. (b) For any function f : A → B, if g : B → A is a function so that go f = IA, then f is one-to-one.
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